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Chen, Lu Mentor: Zhou, Jian Shanghai University, School of Management Email: Chen213.shu.edu.cn

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The object is to minimize the total length of the links inserted into the network. Conditions: Every pair of vertices is connected by exactly one path. n-1 links need to be chosen. No cycle. 2

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Ⅰ Kruskal Algorithm (1956)Kruskal Algorithm (1956) Ⅱ Prim Algorithm (1957)Prim Algorithm (1957) Ⅲ Dijkstra Algorithm (1959)Dijkstra Algorithm (1959) Ⅳ Reverse Delete AlgorithmReverse Delete Algorithm Ⅴ Sollin Algorithm (1961)Sollin Algorithm (1961) 3

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G=( V, E ) w v means vertex. e means edges. w means weight. 4

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1. Sort the edges by weight. 2. Examine each edge in the order of increasing weight. 3. If the edge examined does not create a cycle with the edges in the current forest, it is added to the forest; otherwise, it is discard. 5

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w 12 w 45 w 14 w 25 w 35 w 23 w 13 w 43 √ √ √ × √ × × × 4 1 2 3 5 1 76 8 5 3 2 4 6

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4 1 2 3 5 1 5 3 2 7

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1. Select any node arbitrarily, and then connect it to the nearest distinct node. 2. Identify the unconnected node that is closest to a connected node, and then connect these two nodes. 3. Repeat step 2 until all nodes have been connected. 8

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4 1 2 3 5 1 76 8 5 3 2 4 9

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Ties for the nearest distinct node or the closest unconnected node may be broken arbitrarily, and the algorithm must still yield an optimal solution. 10

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4 1 2 3 5 1 76 8 5 4 2 4 11

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4 1 2 3 5 1 5 2 12 4

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4 1 2 3 5 1 5 2 4 13

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Special Prim algorithm Definitions: S ： Set of the nodes which have been connected by branch. S’ ： The remaining nodes. d(v) ： Distance from note v to S. 14

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Step1: Set N={1, 2,…,n}, and d j =w 1 j. Step2: d j* =min{d j | j∈S’}=w i j* (i∈S, j*∈S’), j* add into S. Step3: d j = min{ d j ， w j*j }, j∈S, return to step 2, until S=N. 15

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4 1 2 3 5 1 76 8 5 3 2 4 16

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1. Select vertex1 as the starting point, S={1}. 2. d 2 =w 12 = 1, d 3 =7, d 4 =3 d j *=min{d 2, d 3, d 4 | j∈S’} =w 12 =w ij* = 1 j*=2,vertex 2 add into S, and S={1,2}. 3. d 4 =min{d 4, w 24 }=min{3, 4}=3 d 3 =min{d 3, w 23 }=min{7, 6}=6 d 5 =min{d 5, w 25 }=min{5, 4}=4 d j *=min{d 4, d 3, d 5 | j∈S’} =w 14 =w ij* =3 j*=4, vertex 4 add into S, and S={1,2,4} 17

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4 1 2 3 5 18 3 （ d 4 =w 14 ） 1 (d 2 =w 12 )

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4. d 3 =min{d 3, w 43 }=min{7, 8}=7 d 5 =min{d 5, w 45 }=min{5, 2}=2 d j *=min{d 4, d 3, d 5 | j∈S’} =w 45 =w ij* =32 j*=5, vertex 5 add into S, and S={1,2,4,5} 5. d 3 =min{d 3, w 53 }=min{7, 5}=5 j*=3, vertex 3 add into S. 6. S={1,2,4,5,3}=N. 19

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4 1 2 3 5 5(w 53 ) 2(w 45 ) 20 3 （ d 4 =w 14 ） 1 (d 2 =w 12 )

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It is the reverse of Kruskal’s algorithm Also called “Tear Cycle Method” 21

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1. Select a spanning tree arbitrarily 2. Add a branch in this tree, then there will be a loop in the graph. Removal the longest one. 3. Repeat step 2 until all the edges have been tested 22

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4 1 2 3 5 1 76 8 5 3 2 4 23

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4 1 2 3 5 1 76 8 5 3 2 4 24

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4 1 2 3 5 1 76 8 5 3 2 4 25

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4 1 2 3 5 1 76 8 5 3 2 4 26

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4 1 2 3 5 1 76 8 5 3 2 4 27

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4 1 2 3 5 1 76 8 5 3 2 4 28

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Original: Borůvka‘s algorithm （ 1926 ） Reason: Because Sollin was the only computer scientist in this list living in an English speaking country, this algorithm is frequently called Sollin's algorithm. 29

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1. Every vertex signify a tree ， and the original graph signify a forest. 2. Chose every vertex’s closet edge. If some vertexes’ closet edge are the same edge, these vertexes merge into a new vertex. 3. Repeat step 2 ， until there is only one tree in the forest. 30

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4 1 2 3 5 1 76 8 5 3 2 4 1e 12 2e 12 3e 35 4e 45 5 e 45 31

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4 1 2 3 5 1 76 85 3 2 4 1,2 e 14 4,5 e 14 3 e 35 32

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4 1 2 3 5 1 76 85 3 2 4 1,2,4,5 e35 3 e35 33

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4 1 2 3 5 1 76 8 5 3 2 4 34

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